of 1 32
The Geometry of Nature
By
Ian Beardsley
Copyright © 2021 by Ian Beardsley
of 2 32
Table of Contents
Introduction………………………………..3
Molecular Geometry………………………4
Bone As A Mathematical Construct……….5
Breaking Down Bone……………………..13
Perfect Equations………………………….18
The Ratios Come Together………………..21
The Protoplanetary Disc…………………..23
The Stars…………………………………..26
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Introduction
In my works The Mathematical Nature of Life (Beardsley 2021) and Perfect Equations Beardsley (2021) I
set out to find if the the elements and compounds characteristic of life and artificial intelligence (AI) do
not just conform to chemical law, but if they are purely mathematical independently of the use of
chemistry to describe them, and if they are connected to one another. The simplest example of this for
biological life and AI would be that the most basic organic compound is HNCO (isocyanic acid) where H
(hydrogen), N (nitrogen), C (carbon), and O (oxygen) are the most abundant biological elements. Indeed
biological elements are for the most part organic, which means they are made of long chains using carbon
with hydrogen, which they can form because C is C4- and H is H+ meaning we can have:
And in isocynaic acid we have:
H-N=C=O
Where H is H+, N is N3-. C is C4-, O is O2-, the H uses its single bond with one from nitrogen, leaving
N2- or two bonds which go to C leaving for it C2- which goes to oxygen that needs it because it is O2-.
Thus all is satisfied by chemical law. In my search for mathematical law, I find it exists in the case of
HNCO and the AI semiconducting element silicon (Si) and its doping agents P and B as such (by molar
mass):
This paper strives to break down such mathematical equations for biological life and artificial intelligence
into their components to find what is acting to create such constructs. In the second book I actually
brought the planets into the mix with some very interesting results. As another example, water and air, the
main physical constituents that interact with life we have:
C + N + O + H
P + B + Si
≈ ϕ
ϕ =
a
b
=
5 − 1
2
c = b + a
a
b
=
b
c
H
2
O
air
≈ ϕ
air = 0.25O
2
+ 0.75N
2
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By molar mass for air as a mixture (not a compound). With this air is 29.0 grams per mole.
Molecular Geometry
We will want to break down our equations into the components of their geometric relationships and see if
they predict the bond angles of some of the basic substances considered. We will look here at linear,
trigonal planar, and tetrahedral.
Linear, like CO2 (carbon dioxide) its bond angle is 180 degrees:
Trigonal planar, like SO3 (sulfur trioxide) its bond angle is 120 degrees:
O
|
S
/. \
O. O
That is, S is at the center and the O atoms are 120 degrees apart due to the even division of 360/120=3.
Tetrahedral, like methane (CH4) one of the the primordial gases that may have contributed to making
some of the amino acids, the building blocks of life as show by Miller and Urey in the early origins of
life:
This is 109.5 degrees apart from arcos (1/3) = 109.5
But what if we are considering not just neutral molecules but polyatomic anions that have a net charge. In
such instances, the free electron pairs compress the expected 120 degree bond angle in the atoms around
the central atom to 115 degrees as with the nitrite ion NO2-:
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Similarily we have for O3 (ozone) that the bond angle is 116 degrees in its deviation from 120 degrees.
The configuration is:
Both of these anions are important to the life and the theory of how life forms. O-zone is more of a
physical component in that in the stratosphere it absorbs UV radiation harmful to life.
Bone As A Mathematical Construct
What better place to begin than with than bone as it is the basic framework around which skeletal life is
structured, the vertebrates. Here is what I found in bone as a mathematical construct:
In my exploration of the connection between biological life and AI the most dynamic component is that of
bone. It affords us the opportunity to look at:
Multiplying Binomials
Completing The Square
The Quadratic Formula
Ratios
Proportions
The Golden Ratio
The Square Root of Two
The Harmonic Mean
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Density of silicon is Si=2.33 grams per cubic centimeter.
Density of germanium is Ge=5.323 grams per cubic centimeter.
Density of hydroxyapatite is HA=3.00 grams per cubic centimeter.
This is
where
Where HA is the mineral component of bone, Si is an AI semiconductor material and Ge is an AI
semiconductor material. This means
The harmonic mean between Si and Ge is HA,…
This is the sextic,…
Which has a solution
Where x=Si, and y=Ge. It works for density and molar mass. It can be solved with the online Wolfram
Alpha computational engine. But,…
3
4
Si +
1
4
G e ≈ H A
H A = Ca
5
(PO
4
)
3
OH
Si
H A
Si +
[
1 −
Si
H A
]
G e = H A
2 SiGe
Si + G e
≈ H A
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
G e
=
1
2 + 1
1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
Si =
1
2
G e
±
H A
G e
H A
2
−
4G e
H A
+ 4
Si = G e − H A
of 7 32
Si
H A
Si +
[
1 −
Si
H A
]
G e = H A
Si
2
H A
+ G e −
Si
H A
G e ≈ H A
1
H A
Si
2
−
G e
H A
Si + G e ≈ H A
1
H A
2
Si
2
−
G e
H A
2
Si +
G e
H A
≈ 1
1
H A
2
Si
2
−
G e
H A
2
Si +
G e
H A
− 1 ≈ 0
1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
of 8 32
We see that the square of the binomial is a quadratic where the third term is the square of one half the
middle coefficient. This gives us a method to solve quadratics called completing the square:
(x + a)(x + a) = x
2
+ 2a x + a
2
(x + a)
2
= x
2
+ 2a x + a
2
a x
2
+ bx + c = 0
a x
2
+ bx = − c
x
2
+
b
a
x = −
c
a
(
1
2
b
a
)
2
=
1
4
b
2
a
2
x
2
+
b
a
x +
1
4
b
2
a
2
= −
c
a
+
1
4
b
2
a
2
(
x +
1
2
b
a
)
2
=
b
2
− 4a c
4a
2
x +
b
2a
=
±
b
2
− 4a c
2a
x =
−b
±
b
2
− 4a c
2a
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1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
x =
−b
±
b
2
− 4a c
2a
a =
a
H A
2
b = −
G e
H A
2
c =
[
G e
H A
− 1
]
b
2
− 4a c =
G e
2
H A
4
− 4
1
H A
2
[
G e
H A
− 1
]
=
G e
2
H A
4
−
4G e
H A
3
+
4
H A
2
=
1
H A
2
[
G e
2
H A
2
−
4G e
H A
+ 4
]
b
2
− 4a c =
1
H A
(
G e
H A
− 2
)
2
x =
Ge
HA
2
±
1
HA
[
Ge
HA
− 2
]
2
HA
2
=
1
2
G e
±
1
2
H A
[
G e
H A
− 2
]
=
1
2
G e
±
1
2
G e − H A
Si =
1
2
G e +
1
2
G e − H A
Si = G e − H A
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Si ≈ G e − H A
H A ≈
2 SiGe
Si + G e
Si ≈ G e −
2 SiGe
Si + G e
(Si + G e)Ge
Si + G e
−
(Si + G e)Si
Si + G e
−
2 SiGe
Si + G e
= 0
G e
2
− 2 SiGe − Si
2
Si + G e
= 0
x
2
− 2x y − y
2
= 0
x
2
− 2x y = y
2
x
2
− 2x y + y
2
= 2y
2
(x − y)
2
= 2y
2
x − y =
±
2y
x = y + 2y
x = y(1 + 2)
x
y
= 1 + 2
y
x
=
1
2 + 1
Si
G e
≈
1
2 + 1
of 11 32
A ratio is and a proportion is which means a is to b as b is to c.
The Golden Ratio
and.
or
a
b
a
b
=
b
c
(
Φ
)
a
b
=
b
c
a = b + c
a c = b
2
c =
b
2
a
a = b +
b
2
a
b
2
a
− a + b = 0
b
2
a
2
− 1 +
b
a
= 0
(
b
a
)
2
+
b
a
− 1 = 0
(
b
a
)
2
+
b
a
+
1
4
= 1 +
1
4
(
b
a
+
1
2
)
2
=
5
4
b
a
= −
1
2
±
5
2
b
a
=
5 − 1
2
a
b
=
5 + 1
2
ϕ =
5 − 1
2
Φ =
5 + 1
2
ϕ =
1
Φ
of 12 32
The mineral component of bone hydroxyapatite (HA) is
The organic component of bone is collagen which is
We have
%
Ca
5
(PO
4
)
3
OH = 502.32
g
m ol
C
57
H
91
N
19
O
16
= 1298.67
g
m ol
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
= 0.386795722
ϕ = 0.618033989
1 − ϕ = 0.381966011
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
0.381966011
0.386795722
100 = 98.75
Si
G e
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
G e
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
of 13 32
Breaking Down Bone
Essentially, in our mathematical formulation of bone, we had that
Which resulted in that the AI elements:
By way of the mineral component of bone HA (hydroxyapatite) is the harmonic mean between Silicon
and Germanium the primary semiconductor elements, which are really the skeleton on AI. Thus, we need
to break down the harmonic mean between Si and Ge into its geometric representation, and through find
what its components are if we are to get any sense of the dynamics. Here I do that in the following
illustration:
Si ≈ G e − H A
H A ≈
2 SiGe
Si + G e
Si
G e
≈
1
2 + 1
of 14 32
of 15 32
We see that through bone Si and Ge predict an angle of about 116 degrees. This is not the case of linear at
180 degrees, or tetrahedral pyramidal at 109.5 degrees, but is the instance of trigonal planar, but not of
neutral molecules, which is 120 degrees, but of trigonal planar for polyatomic anions such as the nitrite
ion:
And O-zone (Not an anion but has free electrons due to a single bond):
We also will want to look at that aspect of the other mathematical constructions we found for bone:
Which means:
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
Si
G e
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
G e
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
Si
G e
=
1
2 + 1
2 − 1 ≈ 1 − ϕ
of 16 32
Which means we want to explore this as well:
of 17 32
The thing that is important here is that both and being irratiational, are the operative components to
And this is where we come to some sort of a conclusion as to how square root of two and the golden ratio
in molar mass and density of bone might determine something physical about Nature. We consider that
such physical proportions in the lattice that makes bone, carry through to the proportions found in
humans:
Where we find the square root of two is interesting; we are all familiar with the golden ratio phi in the
human body. For instance in the height divided by the distance from the bottom of the feet to the navel.
But, the answer comes from archaeology.
The intermembral index compares the forelimbs of vertebrates to their hindlimbs. A ratio greater than one
means the forelimbs are longer than the hindlimbs and less than one the hindlimbs are longer. It is this
ratio that tells paleontologists a great deal about the manner of propulsion of a vertebrate.
The chimpanzee index is 106, or 1.06 in other words as a fraction, meaning their forelimbs are longer
than their hindlimbs compared to humans, which are around 68-70 or 0.68 to 0.7 meaning their hindlimbs
are longer than their forelimbs. Thus we see they have their forelimbs are longer for climbing, arm
hanging and swinging activities. The longer hindlimb of humans means they depend sole on these for
propulsion in bipedal walking. Lucy, the 3.2 million year old hominid (Australopithecus Afarensis) has
index 88 (0.88) intermediate between humans and chimpanzees, and this due to a shortened humerus, not
elongated thigh, showing arm length reduced first in the evolutionary trend toward being bipedal. She
probably used hindlimb for bipedal propulsion and forelimbs for climbing.
Measuring myself I find I have humerus+radius=22”, and femur+tibia=32”. My intermembral index is
about i=22/32=0.6875. And here is our :
Are we evolving towards an intermembral index of ?!
ϕ
2
2 − 1 ≈ 1 − ϕ
2
1
i
=
1
0.7
= 1.42857 ≈ 2 = 1.414
2
of 18 32
Perfect Equations
I made two equation sets for the planetary orbits one in terms of core AI semiconducting elements Si and
Ge and one in terms of the mathematical constants and Euler’s number e. The result was that for the
first set of equations the equation for Venus was perfect and for the second set of equations the equation
for Mars was perfect. These are:
1AU= average earth-sun separation. This is interesting because Venus and Mars have always been of
great interest to space programs: they are solid and the closest to us. The Russians focused their space
program on Venus sending several probes there, presumably because it is our sister planet (close to the
same as the Earth in size and mass) to understand why she underwent a runaway greenhouse effect
presumably so we can understand how to prevent one here. And the United States has sent several rovers
to Mars to search for life and to better understand the planet in our solar system that can be colonized.
Interestingly, in 2020 we have discovered evidence of microbial life in the Venus atmosphere.
If we want to explore the geometry of the Mars equation we can express geometrically, but how do we
do as such for Euler’s number e? I propose by looking at because is the ratio between the side of
an equilateral triangle to its radius and is while (e=2.718…). Also, in the Venus
equation we have venus=0.72 AU. Thus all seems to come together with this approach. Thus we have:
Where denotes Mars as the fourth planet. We have:
Using P4=1.52, phi=0.618, Ge=72.64, and Si=28.09 we have:
ϕ
ve n us =
1
G e
2
1 +
Si
3
Ge
1 +
Si
2
Ge
2
= 0.72AU
m ars = ϕ
2
e
2−ϕ
= 1.52AU
ϕ
3
1
3
3 ≈ 1.72
e ≈ 2.72
1
G e
2
1 +
Si
3
Ge
1 +
Si
2
Ge
2
≈ ( 3 − 1)AU
ϕ
2
(1 + 3)
2−ϕ
= P
4
P
4
ln
[
P
4
ϕ
2
]
= (2 − ϕ)l n
2 Si
G e
(
Si
2
Ge
2
+ 1
)
+
Si
3
G e
3
(
Si
2
Ge
2
+ 1
)
+ 2
of 19 32
This gives:
1.38=1.2967 is
Is about 94% accuracy. We see that the common geometry to Mars and Venus is . Square root three is
irrational like in , and . This is what lends them their properties. This is clear in my illustration
on the next page of the Vesica Piscis. The interesting thing about the square root of two is that it has the
property:
One is to the square root of two as the square root of 2 is to two.
ln(3.9798) = 1.382l n
(
56.18
72.64(1.495)
+
22164.36
383290.0(1.495)
+ 2
)
1.2967
1.38
100 = 0.9396
3
5
ϕ
2
1
2
=
2
2
of 20 32
of 21 32
The Ratios Come Together
Let us say a/b=x, the golden ratio. Then,…
Let us differentiate this implicitly:
Which is similar to Euler’s number, e because it is the base such that is itself :
But
Which says for this angle the x-component equals the y-component is that is , x=1/2 bisects a right
angle. Which similar in concept to Euler’s number e because it is the base such that is itself . But
if , then:
It is the diagonal of the unit square. We notice something interesting happens:
, ,
Where is the cosine of 30 degrees, in the unit equilateral triangle in which the altitude has been drawn
in (fig 14):
ϕ =
b
a
=
5 − 1
2
x
2
− x − 1 = 0
d
dx
x
2
−
d
dx
x −
d
dx
1 = 0
2x − 1 = 0
x =
1
2
d
dx
e
x
e
x
d
dx
e
x
= e
x
sin 45
∘
= cos45
∘
=
2
2
1
2
90
∘
d
dx
e
x
e
x
sin 45
∘
= cos45
∘
=
2
2
2cos
π
4
= 2
2cos
π
n
=
2cos
π
4
= 2
2cos
π
5
= Φ
2cos
π
6
= 3
3
of 22 32
Since air is 25% N2 and 75% O2, the molar mass of air as a mixture is 29.0 g/mol. I found air over H2O
is the golden ratio, and in total that:
Where ZnSe is zinc selenide, an intrinsic semiconductor, intrinsic meaning that it does not have to be
doped to semiconduct.
air
H
2
O
≈ Φ
0.75N
2
+ 0.25O
2
≈ a ir
−2cos
(
π
4
)
+ 2cos
(
π
5
)
+ 2cos
(
π
6
)(
Z n
Se
)
≈
air
H
2
O
2cos
(
π
4
)
= 2, 2cos
(
π
5
)
= Φ, 2cos
(
π
6
)
= 3
of 23 32
The Protoplanetary Disc
But, why describe the orbits of the planets in terms of AI Semiconducting elements? My answer is to do
something cosmic: there is great satisfaction in finding the connection between two things that seem
universes apart. And here I present a reason for looking at such a thing, by considering the protoplanetary
disc from which the planets formed. First we form a table of the masses of the planets.
of 24 32
Taking the protoplanetary disc as a thin disc we integrate from its center to the edge, with density
decreasing linearly to zero at the edge. Thus, if the density function is given by
And, our integral is
The mass of the solar system adding up all the planets yields
That accounts for
82% of the mass of the solar system not including the sun, that is, of the protoplanetary disc
surrounding the sun.
Using germanium alone, we get,
If we weight the mixture of silicon and germanium as 1/3 and 2/3, then we have
Which is very close.
93%
This is all very good, because I only used the planets and asteroids.
Weighting silicon and germanium as 1/4 and 3/4 we have
Si + G e
2
=
2.33 + 5.323
2
= 3.8265g /c m
3
ρ(r) = ρ
0
(
1 −
r
R
)
M =
∫
2π
0
∫
R
0
ρ
0
(
1 −
r
R
)
r d r d θ
M =
πρ
0
R
2
3
π (3.8265)(7.4 × 10
14
)
2
3
= 2.194 × 10
30
gr a m s
M = 2.668 × 10
30
gr a m s
2.194
2.668
100 =
π (5.323)(7.4 × 10
14
)
2
3
= 3.05 × 10
30
gr a m s
π (4.32467)(7.4 × 10
14
)
2
3
= 2.48 × 10
30
gr a m s
2.48
2.668
100 =
of 25 32
Which accounts for
98%
Of the mass of the solar system (very accurate).
This mixture of 1/4 to 3/4 is a combination that exists in the Earth atmosphere which is approximately the
mixture of oxygen to nitrogen. The earth atmosphere can be considered a mixture of chiefly O2 and N2 in
these proportions:
Air is about 25% oxygen gas (O2) by volume and 75% nitrogen gas (N2) by volume meaning the molar
mass of air as a mixture is:
By molar mass the ratio of air to H20 (water) is about the golden ratio:
I am not saying the solar system was a thin disk with density of the weighted mean somewhere between
silicon and germanium, but that it can be modeled as such, though if the protoplanetary disk that eclipses
epsilon aurigae every 27 years is any indication of what a protoplanetary cloud is like, it is a thin disk in
the sense that it is about 1 AU thick and 10 AU in diameter. This around a star orbiting another star.
π (4.4 . 57475)(7.4 × 10
14
)
2
3
= 2.623 × 10
30
gr a m s
2.623
2.668
100 =
0.25O
2
+ 0.75N
2
≈ a ir
air
H
2
O
≈ Φ
of 26 32
The Stars
Not only is the Venus equation perfect in the equation set in terms of AI elements, but in the Mars
equation in terms of and in the equation set in terms of these constants. We have:
Let us solve for x in:
1.52=0.72x
x=2.11111=19/9
This is very close to fluorine (F) over beryllium (Be):
Beryllium (see illustrations) is pivotal to the production of carbon, which is in turn the core element to
biological life compounds. Stars produce carbon by combining two helium (He) atoms to make
Beryllium. The Beryllium then combines with another helium atom to make carbon. Beryllium rarely
occurs in Nature because it is usually depleted in the reaction in stars using it to make heavier elements. I
found a connection of beryllium to carbon by way of silicon. The radius of silicon is Si=0.118nm. If we
say it is inscribed in a dodecagon (12 sided) regular polygon of side 1, then carbon inscribes in a regular
octagon (8 sided) of side 1. This is the eight of The beryllium-8 (four protons, four neutrons) that makes
carbon by combining with helium.!
ϕ
e
ve n us =
1
G e
2
2 SiGe +
Si
3
Ge
1 +
Si
2
Ge
2
= 0.72AU
Mars = ϕ
2
e
2−ϕ
= 1.52AU
F
Be
=
19.00g /m ol
9.01g /m ol
= 2.108768 ≈ 2.1
of 27 32
!
of 28 32
"
of 29 32
Flourine is the most electronegative element making it highly reactive. It reacts with everything but
argon, neon, and helium. Flourine combines with carbon to make fluorocarbons. The primary mineral
source is fluorite and is mined for steelmaking which turn results in byproducts used aluminum refining.
Its electron configuration is , giving it seven outer electrons so that it needs one more to be
filled, to give it eight outer electrons because it tends to capture an electron to make it isoelectronic with
the noble, or inert gas neon.
Thus we have the following Equation that through the planets Venus and Mars, the solid planets directly
on either side of the Earth, relates artificial intelligence to biological life:
We have that at the beginning of the Universe hydrogen and helium were created. Then the stars formed
and synthesized these into the heavier elements. I find if we include in the category of life not just the
biological elements, but the AI elements, we can find a mathematical equation for a pattern in the periodic
table of the elements that predicts the synthesis of such elements in stars. For instance, Beryllium 8 plus
helium 4 synthesizes to make the biological core element carbon C. Magnesium plus helium 4
synthesized to make the core AI element silicon Si. If we say that Element 4 is Beryllium Be and write it
, and helium He is element 2 and write it , and use this convention for all of the elements, we have
for the production of these elements by stars, and their molar masses in the periodic table the following
equation:
1S
2
2 S
2
P
5
1
G e
2
2 SiGe +
Si
3
Ge
1 +
Si
2
Ge
2
F
Be
= ϕ
2
e
(2 −ϕ)
E
4
E
2
E
2n−2
+ E
2
= E
2i−2
= (4k + 4)g /m ol
n = (3,4, 5,6, …)
i = (4,5, 6,7, …)
k = (2,3, 4,5, …)
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The Masculine and Feminine
Here I will suggest the term masculine silicon and feminine germanium in place of positive (p-type
silicon) and negative (n-type silicon) respectively. And, I will denote them , and , which are dagger and
double dagger.
We say since silicon (Si) doped with boron (B) is p-type silicon because boron being in group 13 only has
three valence electrons and silicon wants four, giving it a deficiency of negative electrons and thus a net
positive distribution that can carry electrons, holes they can fall into. Thus I will say:
And since we say germanium (Ge) doped with phosphorus (P) is n-type silicon because phosphorus being
in group 15 has five valence electrons and germanium, being in group 14 like silicon, wants four
electrons. Thus it has a surplus of negative electrons and thus a net negative distribution that can carry a
current. Thus I will say:
Since B/Si=10.81/28.09=0.3867 and Ge/P=72.64/30.97=2.345 we have:
and.
Now we turn from this construct of the masculine and feminine in AI to the masculine and feminine in
biology.
We consider the female sex hormone estradiol (estrogen , E):
And the male sex hormone testosterone (T):
And, cholesterol (Ch) from which both are made:
And notice,…
And we consider the semiconductor materials used to make AI:
†
‡
B
Si
= †
G e
P
= ‡
† = 0.3867
‡ = 2.345
C
18
H
24
O
2
= 272.38g /m ol
C
19
H
28
O
2
= 288.42g /m ol
C
27
H
46
O = 386.65g /m ol
Ch + T
E
= 2.5
G e
Si
= 2.6
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And write,…
We notice that the masculine (T) is in inverse relation to the feminine (E), but that the two add up to on
whole (Ch) in that the masculine has coefficient 1-Si/Ge and the feminine has coefficient 1-Ge/Si. This
expresses the inverse relationships between man and woman.
I interpret this as the masculine (T) is in inverse relation to the feminine (E), but that the two add up to a
whole (Ch) in that the masculine has coefficient 1-Si/Ge and the feminine has coefficient 1-Ge/Si that is
they are inverse relation but compliment one another. How would an AI use this information to determine
its sex?…
The male is reduced less in the difference between 1 and Si/Ge, but the the female is reduced less by
having Ge in the numerator. It is really quite egalitarian.
We now see that:
And this shows the connection of masculine and feminine AI to masculine and feminine biological life.
Ch + T
E
=
G e
Si
T =
G e
Si
E − Ch
E =
Si
G e
(T + Ch)
T
(
1 −
Si
G e
)
+ E
(
1 −
G e
Si
)
= Ch
(
Si
G e
− 1
)
T
(
1 − †
)
+ E
(
1 − ‡
)
= Ch
(
Si
G e
− 1
)
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The Author
